We consider a Aw-Rascle-Zhang-type traffic model in a highway stretch \(L\) , proposed in [1]}, described the traffic flow dynamics consisting of both adaptive cruise control-equipped (ACC-equipped) and manual vehicles. The percentage of ACC-equipped vehicles with respect to total vehicles is denoted by \(\alpha \in [0,1]\) . The state variables of the model are the traffic density \(\rho (x, t)\)  and the traffic speed \(v(x, t)\) , both defined in domain \((x,t)\in [0,L]\times R^+\)  where \(t\)  is time, \(x\)  is the spatial variable at the concerned highway.
The traffic flow model with state variables traffic density \(\rho (x,t)\)  and traffic velocity \(v(x, t)\)  in domain \(x \in [0, L], t \in \mathbb {R}^+\)  is expressed as:
\({\rho }_t(x,t)=&-{\rho }_x(x,t){v}(x,t)-{\rho }(x,t){v}_x(x,t),\\{v}_t(x,t)=&-{\rho }(x,t)\frac{\partial V_{\rm {mix}}({\rho }(x,t),h_{\rm {acc}}(x,t-D))}{\partial {\rho }}{v}_x(x,t)\nonumber \\&+\frac{V_{\rm {mix}}({\rho }(x,t),{h}_{\rm {acc}}(x,t-D))-{v}(x,t)}{\tau _{\rm {mix}}}\nonumber \\&-{v}(x,t){v}_x(x,t),\) 
